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A subset that belongs to the chosen topology on a space.
An open set in a topological space is a subset such that .
Open sets are the basic “observable” sets in topology: they define neighborhoods of points and determine operations like interior and closure. They also control continuity through preimages.
A topological space is an ordered pair where is a set and is a topology on , meaning:
Here denotes the power set of , and the members of are the open sets (whose complements are the closed sets). Many standard constructions—such as the subspace topology, product topology, and quotient topology—produce new topological spaces from existing ones.
A subset of a set is a set such that every element of is an element of :
A neighborhood of a point in a topological space is a subset such that there exists an open set with .
The interior of a subset in a topological space is
The closure of a subset in a topological space is
A continuous map between topological spaces and is a function such that for every open set , the preimage is open in .